Exploring expected values of topological indices of random cyclodecane chains for chemical insights

Chemical graph theory has made a significant contribution to understand the chemical compound properties in the modern era of chemical science. At present, calculation of the topological indices is one of most important area of research in the field of chemical graph theory. Cyclodecane is a cyclic hydrocarbon with the chemical formula \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{10}H_{20}$$\end{document}C10H20. It consists of a ring of ten carbon atoms bonded together in a cyclical structure. Cyclodecane chains can be part of larger molecules or polymers, where multiple cyclodecane rings are connected together. These molecules can have various applications in chemistry, materials science, and pharmaceuticals. This article aims to determine expected values of some connectivity based topological indices of random cyclodecane chains, containing saturated hydrocarbons with at least two rings. It also compares these descriptors using explicit formulae, numerical tables and present graphical profiles of these comparisons.

theory, the readers can see 10 .The Randić index 18 , first introduced by Milan Randić in 1975, measures molecular branching of chemical compounds in graph theory.The mathematical formula of Randić index is It is useful in quantitative structure-activity relationship (QSAR) studies in chemistry, correlated with properties like boiling points, enthalpies, and molecular weights.It captures information about molecular structure branching and connectivity, making it a valuable tool in chemical graph theory and molecular graph analysis.Details on these applications can be found in the books [19][20][21] .The General Randić index 22 , also known as the General Randić connectivity index, is an extension of the Randić index, focusing on the molecular branching of chemical compounds.The general Randic index of a graph ϒ is defined as The Atom-Bond Connectivity (ABC) index 23 is a mathematical tool in chemistry used to analyze the structure of molecules, measure their complexity.The mathematical formula of ABC index is It is used in Quantitative Structure-Activity Relationship studies, molecular descriptors and cheminformatics to study interactions, describe molecules and analyze chemical data 24,25 .The Atom-Bond Sum Connectivity (ABS) index 26 is a topological index used in chemical graph theory to quantify the molecular structure of chemical compounds.It provides a numerical descriptor of molecular structure, useful in computational chemistry, quantitative structure-activity relationship studies, and other areas.It captures information about atom connectivity and bond types, enabling correlation with molecular properties and activities.It is defined as The geometric arithmetic index 27 , which combines geometric and arithmetic mean values of molecular graph properties, helps chemists understand molecule structural characteristics and predict their behavior in chemical processes or biological activities.The geometric arithmetic index of a graph ϒ has the mathematical formula The paper is structured as follows: In Section "Materials and methods", we discuss the 2D and 3D models of cyclodecanes and their properties.We explain the construction of random cyclodecane chains, and we have obtained general formulas for some connectivity-based topological indices.In Section "Main results and discussions", we compute explicit expressions for the connectivity-based topological indices of random cyclodecane chains.The expressions for the expected values of these topological descriptors are obtained for some special cases.An analytical comparison between the expected values of these topological descriptors is presented in Section "Comparison between the expected values of topological descriptors".Finally, the conclusion section summarizes the article.

Materials and methods
Cyclodecane is a ten-carbon ring with ten membered rings, with two possible isomers, cis-cyclodecane and trans-cyclodecane (see Fig. 1).It undergoes Bergmann cyclization to produce diradical products that inhibit cell replication and interact with DNA.The 2D chemical structure of cyclodecane, also known as the skeletal formula, is the standard notation for organic molecules.Carbon atoms are located at the corner(s) and hydrogen atoms are not indicated.Each carbon atom is associated with enough hydrogen atoms to form four bonds.The 3D chemical structure image of cyclodecane uses a ball-and-stick model, displaying atom positions and bonds.The radius of spheres is smaller than rod lengths, allowing for a clearer view of atoms and bonds.In comparison to typical polymers, cyclodecane-based monomers enable polymer synthesis, resulting in unique polymers with cyclodecane-containing characteristics.Cyclodecane may impact the crystal structure of certain compounds, particularly those with coordination complexes or molecular assemblies, affecting the packing arrangement and overall properties of the crystal lattice.The chemical structure of a molecule contains the arrangement of its atoms and the bonds that hold them together.Cyclocodecane has 30 bonds, including 10 non-hydrogen bonds and 1 ten-numbered ring.The 2D and 3D models of cyclodecane chains are depicted in Fig. 1.The structure of the cyclodecane chain is chemical as well.Some of the characteristics of cyclodecane chains are: Molecular Weight 140.Researcher have focused on hydrocarbons and their derivatives because of their simple structure have two components carbon and hydrogen.Numerous kinds of hydrocarbon derivatives can be obtained by substituting their molecular hydrogen atoms with various other atomic groups.Plants contains a significant amount of (1) .
precious hydrocarbons and some of these hydrocarbons properties are important in the production of chemical raw material and fuel.A cycloalkane with the chemical formula C 10 H 20 is cyclodecane.When an edge is used to join the two or more decagons then it is known as cyclodecane chain.A random cyclodecane of length k is a chain containing k decagons which are connected to each other by edge in a random way.We use the notation CDC k to denote a random cyclodecane chain containing k decagons.Figure 2 shows the unique cyclodecane CDC k for k = 1, 2 .There are five possible ways to connect a terminal decagon with the cyclodecane chain CDC k−1 with probability δ 1 , δ 2 , δ 3 , δ 4 , and δ 5 = 1 − δ 1 − δ 2 − δ 3 − δ 4 respectively.A random selection is made from one of the five possibilities at each step (q = 3, 4, 5, ..., k) : For k = 3 , we have five different possible cyclodecane chains (see Fig. 3).The five different configurations of cyclodecane chains CDC 1 k+1 , CDC 2 k+1 , CDC 3 k+1 , CDC 4 k+1 and CDC 5 k+1 are shown in Fig. 4. For results on the expected values of different topological indices of random structures see [28][29][30][31][32][33][34][35][36][37][38] .
In this section, we compute the expected values of geometric-arithmetic index, atom-bound connectivity index, atom-bound-sum connectivity index, Randić index and general Randić index for CDC k chain having k decagons.Consider CDC k to be the cyclodecane chain formed from CDC k−1 , as illustrated in Fig. 4. We use the notation υ ij to denote the number of edges of CDC k whose end vertices have degree i and j respectively.The structure of the chain CDC k clearly shows that it comprises only (2, 2), (2, 3), and (3, 3) type edges.To calculate these indices for the chain CDC k , we need to find the edges of the type υ 22 (CDC k ) , υ 23 (CDC k ) and υ 33 (CDC k ) .Using this information, Eqs. ( 1), ( 2), ( 3), ( 4) and ( 5) can be written as:

Main results and discussions
For k ≥ 3 , the cyclodecane chain CDC k is a random structure.It follows GA(CDC k ) , ABC(CDC k ) , ABS(CDC k ) , R(CDC k ) and GR(CDC k ) are random variables.We use the notaions Using these values in Eq. ( 6), we get Using these values in Eq. ( 6), we get Using these values in Eq. ( 6), we get Using these values in Eq. ( 6), we get Using these values in Eq. ( 6), we get Thus, we have Using these values in Eq. ( 7), we get Using these values in Eq. ( 7), we get Using these values in Eq. ( 7), we get

ABC(CDC
Using these values in Eq. ( 8), we get Using these values in Eq. ( 9), we get Using these values in Eq. ( 9), we get Using these values in Eq. ( 9), we get Using these values in Eq. ( 9), we get Using these values in Eq. ( 9), we get Thus, we have Using these values in Eq. ( 10), we get Using these values in Eq. ( 10), we get Using these values in Eq. ( 10), we get Using these values in Eq. ( 10), we get Using these values in Eq. ( 10), we get www.nature.com/scientificreports/Thus, we have Finally, solving the the recurrence relation by using the initial condition E(CDC 2 ) = 7(4 γ ) + 2(6 γ ) + 2(9 γ ) , we get We now focus on the unique cyclodecane chains CF k , CG k , CH k , CI k and CJ k (see Fig. 5).These chains can be obtained from CDC k as special cases by taking the value of one of the probability δ i = 1 and the remaining probabilities 0 at each step, where i = 1, 2, . . ., 5 .We use Theorems 1, 2, 3 and 4 to calculate the topological indices for these five specific chains.
Corollary 6 Let k ≥ 2 , then we have

Comparison between the expected values of topological descriptors
In this section we compare the expected values for the Randić, general Randić, atom-bound connectivity, atombound-sum connectivity and geometric-arithmetic indices for random cyclodecane chain having same probabilities.Tables 1, 2, 3, and 4 provides the numerical values of the expected values of these topological descriptors for different values of the probability function δ 1 .It is easy to observe that the value of geometric-arithmetic index is always greater than the other topological descriptors in all the cases.The comparison of the expected values of these topological descriptors can be seen in Figs. 6 and 7. Now, we give an analytical proofs for the comparison of the expected values of the considered topological descriptors.
Proof The statement is true for k = 2 .Now, we prove that the statement is true for k > 2 .By using Theorem 1 and 4, we have 27 g/mol, Melting Point 10.0 • C , Boiling Point 202.0 • C , Health Risk 0.33 mg/L, Water Solubility 25 • C and Vapour Pressure 0.56 mmHg.

Figure 3 .
Figure 3.The five types of cyclodecane chain for k = 3.

Finally, solving the
the recurrence relation by using the initial condition E(CDC 2 ) = 20.9192 ,we get Theorem 2 Let k ≥ 2 , then the expected value of the atom-bound connectivity index of CDC k is Proof For k = 2 , we get E ABC (CDC 2 ) = 14.81 which is indeed true.Let k ≥ 3 , then there are five possibilities.a

Figure 6 .Theorem 9
Figure 6.Graphical comparison between the expected values of topological indices.